- Open Access
Magneto-optical response of five-septuple-layer in spin-flip states
Phys. Rev. B 114, 165413 – Published 15 September, 2026
DOI: https://doi.org/10.1103/33km-cp5r
Abstract
Magneto-optical (MO) effects like Kerr and Faraday rotations provide a direct probe of topological order in thin films of the magnetic topological insulator (TI) (MBT). Motivated by recent experimental studies of spin-flip/flop transitions in MBT thin films, we investigate the interplay between interlayer spin configurations, topological order, and MO response in five septuple-layer (5-SL) MBT using first-principles calculations and a simplified coupled Dirac cone model. Our results reveal that, despite possessing a nonzero out-of-plane magnetization, 5-SL MBT thin films can be either TIs or topologically trivial insulators depending on the relative spin orientations of the top and bottom SLs. We evaluate the Faraday and Kerr rotation angles using tight-binding models derived from ab initio calculations and by comparing our results with those of a simplified coupled Dirac cone model to clarify the macroscopic mechanisms underlying the MO response of spin-flip states. These theoretical findings highlight the tunability of topological and MO properties in MBT thin films and provide microscopic insight into the emergence of complex topological order in layered antiferromagnetic materials.
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References (58)
- R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, et al., Terahertz response and colossal Kerr rotation from the surface states of the topological insulator , Phys. Rev. Lett. 108, 087403 (2012).
- L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N. Armitage, Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator, Science 354, 1124 (2016).
- K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state, Nat. Commun. 7, 12245 (2016).
- M. Mogi, Y. Okamura, M. Kawamura, R. Yoshimi, K. Yasuda, A. Tsukazaki, K. Takahashi, T. Morimoto, N. Nagaosa, M. Kawasaki, et al., Experimental signature of the parity anomaly in a semi-magnetic topological insulator, Nat. Phys. 18, 390 (2022).
- M. Lang, M. Montazeri, M. C. Onbasli, X. Kou, Y. Fan, P. Upadhyaya, K. Yao, F. Liu, Y. Jiang, W. Jiang, et al., Proximity induced high-temperature magnetic order in topological insulator-ferrimagnetic insulator heterostructure, Nano Lett. 14, 3459 (2014).
- X. Che, K. Murata, L. Pan, Q. L. He, G. Yu, Q. Shao, G. Yin, P. Deng, Y. Fan, B. Ma, et al., Proximity-induced magnetic order in a transferred topological insulator thin film on a magnetic insulator, ACS Nano 12, 5042 (2018).
- D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Topological axion states in the magnetic insulator with the quantized magnetoelectric effect, Phys. Rev. Lett. 122, 206401 (2019).
- J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, Intrinsic magnetic topological insulators in van der Waals layered -family materials, Sci. Adv. 5, eaaw5685 (2019).
- M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M. Echenique, A. Arnau, and E. V. Chulkov, Unique thickness-dependent properties of the van der Waals interlayer antiferromagnet films, Phys. Rev. Lett. 122, 107202 (2019).
- Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Quantum anomalous Hall effect in intrinsic magnetic topological insulator , Science 367, 895 (2020).
- C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. Zhang, and Y. Wang, Robust axion insulator and Chern insulator phases in a two-dimensional antiferromagnetic topological insulator, Nat. Mater. 19, 522 (2020).
- C. Liu, Y. Wang, M. Yang, J. Mao, H. Li, Y. Li, J. Li, H. Zhu, J. Wang, L. Li, et al., Magnetic-field-induced robust zero Hall plateau state in Chern insulator, Nat. Commun. 12, 4647 (2021).
- W. Lin, Y. Feng, Y. Wang, J. Zhu, Z. Lian, H. Zhang, H. Li, Y. Wu, C. Liu, Y. Wang, et al., Direct visualization of edge state in even-layer at zero magnetic field, Nat. Commun. 13, 7714 (2022).
- Y.-F. Zhao, L.-J. Zhou, F. Wang, G. Wang, T. Song, D. Ovchinnikov, H. Yi, R. Mei, K. Wang, M. H. Chan, et al., Even–odd layer-dependent anomalous Hall effect in topological magnet thin films, Nano Lett. 21, 7691 (2021).
- D. Ovchinnikov, X. Huang, Z. Lin, Z. Fei, J. Cai, T. Song, M. He, Q. Jiang, C. Wang, H. Li, et al., Intertwined topological and magnetic orders in atomically thin Chern insulator , Nano Lett. 21, 2544 (2021).
- F. Lüpke, A. D. Pham, Y.-F. Zhao, L.-J. Zhou, W. Lu, E. Briggs, J. Bernholc, M. Kolmer, J. Teeter, W. Ko, et al., Local manifestations of thickness-dependent topology and edge states in the topological magnet , Phys. Rev. B 105, 035423 (2022).
- Y. Li, Y. Wang, Z. Lian, H. Li, Z. Gao, L. Xu, H. Wang, R. Lu, L. Li, Y. Feng, et al., Fabrication-induced even-odd discrepancy of magnetotransport in few-layer , Nat. Commun. 15, 3399 (2024).
- B. Chen, X. Liu, Y. Li, H. Tay, T. Taniguchi, K. Watanabe, M. H. Chan, J. Yan, F. Song, R. Cheng, et al., Even–odd layer-dependent exchange bias effect in Chern insulator devices, Nano Lett. 24, 8320 (2024).
- J.-X. Qiu, B. Ghosh, J. Schütte-Engel, T. Qian, M. Smith, Y.-T. Yao, J. Ahn, Y.-F. Liu, A. Gao, C. Tzschaschel, et al., Observation of the axion quasiparticle in 2D , Nature (London) 641, 62 (2025).
- X. Han, A.-H. Chen, M. Brahlek, and L. Wu, Quantized magneto-terahertz effects in the antiferromagnetic topological insulator thin films, arXiv:2503.13651.
- J.-X. Qiu, C. Tzschaschel, J. Ahn, A. Gao, H. Li, X.-Y. Zhang, B. Ghosh, C. Hu, Y.-X. Wang, Y.-F. Liu, et al., Axion optical induction of antiferromagnetic order, Nat. Mater. 22, 583 (2023).
- F. M. Bartram, M. Li, L. Liu, Z. Xu, Y. Wang, M. Che, H. Li, Y. Wu, Y. Xu, J. Zhang, et al., Real-time observation of magnetization and magnon dynamics in a two-dimensional topological antiferromagnet , Sci. Bull. 68, 2734 (2023).
- P. M. Sass, J. Kim, D. Vanderbilt, J. Yan, and W. Wu, Robust A-type order and spin-flop transition on the surface of the antiferromagnetic topological insulator , Phys. Rev. Lett. 125, 037201 (2020).
- G. Qian, M. Shi, H. Chen, S. Zhu, J. Hu, Z. Huang, Y. Huang, X.-H. Chen, and H.-J. Gao, Spin-flop transition and Zeeman effect of defect-localized bound states in the antiferromagnetic topological insulator , Nano Res. 16, 1101 (2023).
- Z. Lian, Y. Wang, Y. Wang, W.-H. Dong, Y. Feng, Z. Dong, M. Ma, S. Yang, L. Xu, Y. Li, et al., Antiferromagnetic quantum anomalous Hall effect under spin flips and flops, Nature (London) 641, 70 (2025).
- W.-K. Tse and A. MacDonald, Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems, Phys. Rev. B 84, 205327 (2011).
- C. Lei and A. H. MacDonald, Kerr, Faraday, and magnetoelectric effects in thin films, Phys. Rev. B 108, 125424 (2023).
- C. Lei, O. Heinonen, A. H. MacDonald, and R. J. McQueeney, Metamagnetism of few-layer topological antiferromagnets, Phys. Rev. Mater. 5, 064201 (2021).
- C. Lei, T. V. Trevisan, O. Heinonen, R. J. McQueeney, and A. H. MacDonald, Quantum anomalous Hall effect in perfectly compensated collinear antiferromagnetic thin films, Phys. Rev. B 106, 195433 (2022).
- P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
- G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
- G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
- S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132, 154104 (2010).
- S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study, Phys. Rev. B 57, 1505 (1998).
- N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
- A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of : A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 185, 2309 (2014).
- G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, et al., as a community code: New features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
- Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, : An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018).
- S. S. Tsirkin, High performance Wannier interpolation of Berry curvature and related quantities with code, npj Comput. Mater. 7, 33 (2021).
- R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
- D. Greenwood, The Boltzmann equation in the theory of electrical conduction in metals, Proc. Phys. Soc. 71, 585 (1958).
- H. Padmanabhan, M. Poore, P. K. Kim, N. Z. Koocher, V. A. Stoica, D. Puggioni, H. Wang, X. Shen, A. H. Reid, M. Gu, et al., Interlayer magnetophononic coupling in , Nat. Commun. 13, 1929 (2022).
- J.-Q. Yan, S. Okamoto, M. A. McGuire, A. F. May, R. J. McQueeney, and B. C. Sales, Evolution of structural, magnetic, and transport properties in , Phys. Rev. B 100, 104409 (2019).
- Y. Jie, X. Cai, Y. Lin, K. Watanabe, T. Taniguchi, J. Yan, D. Ovchinnikov, and A. Avsar, Reconfigurable magnetotransport in via gate and magnetic field tuning, Adv. Mater. 37, e10734 (2025).
- C. Lei, S. Chen, and A. H. MacDonald, Magnetized topological insulator multilayers, Proc. Natl. Acad. Sci. USA 117, 27224 (2020).
- A. Alfonsov, K. Mehlawat, A. Zeugner, A. Isaeva, B. Büchner, and V. Kataev, Magnetic-field tuning of the spin dynamics in the magnetic topological insulators , Phys. Rev. B 104, 195139 (2021).
- W. Qi, F. Fei, Z. Zhang, B. Chen, H. Xie, B. Wei, S. Zhang, and F. Song, Exchange bias in heterostructures combining magnetic topological insulator and metallic ferromagnet , Appl. Phys. Lett. 125, 023101 (2024).
- Y. Lai, L. Ke, J. Yan, R. D. McDonald, and R. J. McQueeney, Defect-driven ferrimagnetism and hidden magnetization in , Phys. Rev. B 103, 184429 (2021).
- M. Garnica, M. M. Otrokov, P. C. Aguilar, I. I. Klimovskikh, D. Estyunin, Z. S. Aliev, I. R. Amiraslanov, N. A. Abdullayev, V. N. Zverev, M. B. Babanly, et al., Native point defects and their implications for the Dirac point gap at (0001), npj Quantum Mater. 7, 7 (2022).
- M. Liu, C. Lei, H. Kim, Y. Li, L. Frammolino, J. Yan, A. H. Macdonald, and C.-K. Shih, Visualizing the interplay of Dirac mass gap and magnetism at nanoscale in intrinsic magnetic topological insulators, Proc. Natl. Acad. Sci. USA 119, e2207681119 (2022).
- H. Tan and B. Yan, Distinct magnetic gaps between antiferromagnetic and ferromagnetic orders driven by surface defects in the topological magnet , Phys. Rev. Lett. 130, 126702 (2023).
- S. Sattar, D. Hedman and C. M. Canali, Surface reconstructions in thin films of magnetic topological insulator , Phys. Rev. Res. 7, 023024 (2025).
- S.-K. Bac, K. Koller, F. Lux, J. Wang, L. Riney, K. Borisiak, W. Powers, M. Zhukovskyi, T. Orlova, M. Dobrowolska, et al., Topological response of the anomalous Hall effect in due to magnetic canting, npj Quantum Mater. 7, 46 (2022).
- See Supplemental Material at https://link.aps.org/supplemental/10.1103/33km-cp5r for derivation of the Kerr angle formula, details on the components of optical conductivity tensor, MO properties of 5-SL having higher magnetization, layer-dependent Chern contributions for different spin-flip configurations of a 7-SL MBT film, and the effect of broadening parameters, which includes Refs. [26, 27].
- M. Mogi, M. Kawamura, A. Tsukazaki, R. Yoshimi, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Tailoring tricolor structure of magnetic topological insulator for robust axion insulator, Sci. Adv. 3, eaao1669 (2017).
- V. Sunko, S. Ahsanullah, V. Jain, S. Weber, S. Kumaran, J. Yan, J. Orenstein, and D. Ovchinnikov, Magneto-optical Kerr effect in an A-type antiferromagnet, Nat. Commun. 17, 7364 (2026).
- E. I. Organick, A Primer (Addison Wesley Longman Publishing Co., Inc., Boston, 1963).
- This can be understood by looking at the analytical expressions of the conductivity tensor derived from the coupled Dirac cone model, see Eqs. (S10)–(S13) in the Supplemental Material [54]. Indeed, both and contain logarithmic and inverse tangential terms, generating sharp peaks and steps. Furthermore, the mass term () in these expressions defines the depth of the peak, followed by a smooth logarithmic tail. The logarithmic function itself tends to zero, but as the number of transitions increases, the logarithmic tail is truncated. Each transition introduces a more negative value into , leading to an overall linear decrease of the function, the slope of which becomes steeper by adding transitions.